Posted
by
timothy
on Sunday August 28, 2011 @01:52PM
from the braver-man-than-I-am dept.

First time accepted submitter sjwaste writes "Slashdot posts a fair number of physics stories. Many of us, myself included, don't have the background to understand them. So I'd like to ask the Slashdot math/physics community to construct a curriculum that gets me, an average college grad with two semesters of chemistry, one of calculus, and maybe 2-3 applied statistics courses, all the way to understanding the mathematics of general relativity. What would I need to learn, in what order, and what texts should I use? Before I get killed here, I know this isn't a weekend project, but it seems like it could be fun to do in my spare time for the next ... decade."

Madness indeed. I got quite deep into physics and calculus at university and hit a brick wall with multivariable calculus. I believe that you'll need the multivariable calculus skills in order to get any reasonable grip on general relativity. You'll also need a strong physics background: force, mass, acceleration, rotary motion, etc. Having read Einstein's book on special relativity, I'd definitely say start there. It's pretty clear and amazingly intuitive. The Feynman lectures on physics are probably the best physics textbook ever. I wonder too if you might find a class on it online -- maybe Harvard or MIT:http://ocw.mit.edu/courses/physics/8-962-general-relativity-spring-2006/ [mit.edu]

The actual math needed to understand the basics of relativity[1] is actually quite simple. If you've had calculus, you have more than you need.

The hard part is wrapping your brain around the concepts and the fact that the rules you use to interact with the world around you are a subset of the rules of the universe.

A book I have recommended several times for people who want to start learning about physics is 'Asimov on Physics'. Dr. Asimov was a master of explaining difficult science in a way that laymen could understand.

[1] Going beyond the basic, or getting into odder corners of general relativity, is another matter.

General relativity? I'm doubtful. To even phrase it you need to know something about Reimannian manifolds (see http://en.wikipedia.org/wiki/Einstein_field_equations#Mathematical_form), which is way beyond something you'd see in standard calculus or even most undergrad math programs. Sure there are a lot of intuitive concepts that can be expressed without the math, but unless you understand the math, it's hard to see how things like frame-dragging are predicted by the theory.

I didn't follow Bra-Ket notation at all until I read up on the history of it. For me, it helped a lot to know Dirac invented it, and that it was needed because it applied to Hilbert spaces, and that Hilbert developed that concept a few years before Dirac got started, and that John von Neumann was the guy who actually named Hilbert's concept "Hilbert Spaces". Why did those things matter?1. Hilbert was discussing infinities, and he was familiar with Cantor's work (and liked it) so he was using the modern definition of infinities (plural), where there are multiple trans-finites possible. His math was meant to cover all that, and the use of it for QM was a limited case. Some events can be described using a quite limited number of spatial dimensions and the results will be understandable with a little calculus or even trig if you just understand how to take the notation used and put it into actual equations. For example, there's a Hilbert for a three dimensional Euclidean space. Other (particularly in QM) events need many spatial dimensions to describe, sometimes even an infinite number.2. The Ket part of the notation is about those vectors in a Hilbert space. You could represent that Euclidean space I mentioned with just a Ket notation, for example. Since Hilbert spaces can have either a finite number of dimensions or an infinite number, and can entail complex numbers, the Bra part becomes needed when the Hilbert space has complex numbers involved. The Bra and Ket together are a short way of writing a formula for a complex conjugate, and the whole can be expressed just as a complex number. These can be mathematically manipulated by partial differential equations. Any person with a fair knowledge of Linear Algebra can derive information from them, secure that the treatment is mathematically both complete and rigorous. That seems to be the real point of the notation, it gets results into a form where the rest of the process uses math that's regarded as rock solid.3. Dirac invented other math for areas where the completeness condition of all Hilbert Spaces didn't apply. He called some of these "rigged Hilbert Spaces" . He proved people could use the Bra-Ket system and similar operations to describe those QM events, but the results won't technically be proven to be correct in an absolute mathematical sense. many working physicists do it anyway.4. People tend to refer to Feynman for a good source to understand all this and not mention von Neumann as much, but it looks like von N. was historically quite involved in it. Maybe some of what he wrote on QM could clarify Bra-Ket notation better for you than the standard modern textbooks.

You can understand the outcomes without the math. You can NOT understand the "why" without the math. I'll leave it as an at home exercise whether those people you know actually understand general relativity, or just know the implications of it.

As Kelvin once said, you haven't really understood anything till you can put numbers on it. Intuitively we understand that a baseball with follow a parabolic path when it's moving in a gravitational field. But unless you can calculate the speed, the angle and the other vairables, the understanding is imperfect.